Does the orientation of the adjacency matrix of a digraph matter?

Question

Consider a digraph $G$ where $U=\{1,2,3,4\}$, $V=\{(1,2),(3,4),(3,2)\}$, then the adjacency matrix is

$$M=\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

but is it $M$ or $M^T$?

Answer 1

In an adjacency matrix, $a_{i,j} = 1$ if and only if there is an edge from vertex $i$ to vertex $j$. $i$ is the row and $j$ is the column.

Consequently, if $(1,2)$ is an edge, $a_{1,2} = 1$, which is the case for your $M$.